# Uniform one-factorisations of the complete bipartite graph

*Last modified: Sept. 5, 1996*

*Joint work with Gillian Nonay with significant contributions from Dan
Archdeacon and Gordon Royle. Thanks to Cindy Lau for help in setting up
these Web pages.*
*These pages are still under construction. In particular, the table is
incomplete; some known examples remain to be entered.*

Select items from the table at left.

### Overview

The table at left lists, for *2<=n<=11*, all possible partitions
of the integer *n* into parts of size larger than one. For example,
the partition of *8* into *3+3+2* is encoded
as *[3,3,2]*. For each such partition *[k*_{1},k_{2},
...,k_{s}], we ask **"Is there a one-factorisation of
the complete bipartite graph ***K*_{n,n}
having the property that the union of any two distinct one-factors is
isomorphic to a *2*-regular graph having components of size
*2k*_{1}, 2k_{2},..., 2k_{1}?" The
table indicates *"Yes"* if an example is known, *"No"*
if it is known that no example exists, and *"?"* in the remaining
cases. If one clicks on any link in the table, an example, construction,
or non-existence argument appears in the frame above.